President Barack Obama today set aside his reticence on the immigration debate, gathering a disparate group of bill supporters around him for only his second public event to tout the overhaul.

Among the 20 people flanking him in the East Room were U.S. Chamber of Commerce President Thomas J. Donohue, AFL-CIO President Richard Trumka, Democratic Mayor Julian Castro of San Antonio and former Commerce Secretary Carlos Gutierrez, a Republican.

The pep talk came four hours before the first Senate test vote on the legislation, another sign that, at least until the amendment process gets started, the bill has enough bipartisan support to comfortably clear the 60-vote threshold for oppositional filibusters.

Were the outcome of the 2:15 p.m. vote in doubt, the president would have almost surely continued to keep his distance from the debate. Rather than use his powers of persuasion to prod the bill along, Obama has taken the opposite approach to advance the one top-flight domestic priority of his second term: He's staying out of the limelight, hoping that doing so will make it less dangerous for a good number of wavering Republicans to get behind the bill.

“The moment is now,’’ Obama said, for legislation he described as filled with appropriate compromises that would make few constituencies entirely happy. “If you’re serious about really fixing the system, then this is the vehicle to do it."

The bill combines bolstered border security with a new 15-year process for the millions of undocumented immigrants who want to become citizens. The president touted both the toughness of the border security provisions and the toughness of the requirements for illegal immigrants who want to become citizens.

"Yes, they broke the rules," he said. "They didn't wait their turn. They shouldn't be let off easily, they shouldn't be allowed to game the system."

Also today, the Service Employees International Union announced it would spend more than $1 million in the next three weeks on a TV advertising campaign urging Senate passage of the bill.

It seems certain that at least 52 of the 54 Senate Democrats will vote in favor of beginning substantive deliberations on the bill. (The only holdouts might come from the roster of four senators seeking re-election next year in solid red states: Mark Pryor of Arkansas, Mary L. Landrieu of Louisiana, Kay Hagan of North Carolina and Mark Begich of Alaska; they look likelier to stray from the party line on amendments later on.)

Nine Republicans have already committed publicly to voting to get the debate started: John McCain, Lindsey Graham, Jeff Flake and Marco Rubio from the “gang of eight”; the Judiciary Committee’s senior Republicans, Charles E. Grassley and Orrin G. Hatch; the top two leaders of the minority, Mitch McConnell and John Cornyn; and New Hampshire's Kelly Ayotte.

Some senators with deep misgivings about the bill will still vote to begin the debate so as not to be labeled obstructionist.

Dozens of amendments may see votes in the next three weeks, with a vote on passage expected by June 28. The larger the majority for passage, the more likely it is that House GOP leaders will green-light debate on a similar bill.

The central narrative of the Senate debate is that Republicans such as Florida's Rubio will be pushing amendments to tighten border security and make sure the path to citizenship remains arduous, just enough to make the legislation more palatable to conservatives but not so much as to cause Democrats to fall away.

Conservative Republican critics of the path to citizenship will be pushing border security amendments seen as poison pills for the bill. Liberal Democrats are considering proposals — on gun ownership by would-be citizens and the rights of gay immigrants, most notably — that if adopted could cause the bipartisan coalition to fall apart.

To that end, Rubio this morning proposed a change he said would close a loophole undermining the bill’s English proficiency requirement for people obtaining a green card, which signifies permanent legal residency.

“Since the bill was introduced two months ago, the open and transparent process it has undergone has elicited constructive criticisms to improve it,” he said. “This is one of the bill’s shortcomings that came to light, which we can now fix.”